Sophie, I think that the inequalities from this Lecture 27 are the "correct" ones. A nice way to remember which way the unit condition should go is to note that there the greater or equal side is of the form \\(f(-)\\), both in \\(f(x) \otimes f(x') \leq f(x\otimes x')\\) and in \\(I \leq f(I)\\). These two inequalities are like the multiplication and the unit of a monoid: in the case of the multiplication, you have *two* things going in and *one* thing coming out, which corresponds to two \\(f\\)'s on the left and one on the right. The unit of a monoid is just a constant, which means that it has *no* thing going in and again *one* thing coming out; just like the \\(f\\)'s in \\(I\leq f(I)\\).

In case that all this sounds a bit mysterious or confusing to you now, let me just say that it will become clear further along the course! (Assuming that lax monoidal functors will be covered.)

In case that all this sounds a bit mysterious or confusing to you now, let me just say that it will become clear further along the course! (Assuming that lax monoidal functors will be covered.)